Supersymmetry, homology with twisted coefficients and n-dimensional knots

TL;DR

This paper introduces a supersymmetric quantum system for n-dimensional knots, defining topological invariants called Witten indexes that relate to classical knot invariants like Alexander polynomials.

Contribution

It constructs explicit supersymmetric quantum invariants for n-dimensional knots, linking them to homology with twisted coefficients and classical knot invariants.

Findings

Witten indexes are topological invariants for n-dimensional knots.

Non-zero Witten indexes imply nontrivial Alexander polynomials.

Witten indexes relate to homology with twisted coefficients.

Abstract

Let $n$ be any natural number. Let $K$ be any $n$-dimensional knot in $S^{n+2}$. We define a supersymmetric quantum system for $K$ with the following properties. We firstly construct a set of functional spaces (spaces of fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Thus we obtain a set of the Witten indexes for $K$. Our Witten indexes are topological invariants for $n$-dimensional knots. Our Witten indexes are not zero in general. If $K$ is equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten indexes restrict the Alexander polynomials of $n$-knots. If one of our Witten indexes for an $n$-knot $K$ is nonzero, then one of the Alexander polynomials of $K$ is nontrivial. Our Witten indexes are connected with homology with twisted coefficients. Roughly speaking, our Witten indexes have path integral representation by using a usual manner of supersymmetric theory.